260 REPORTS ON THE STATE OF SCIENCE, ETC. 
where G”, G’Y, etc. are the coefficients of the even order differences in Gauss’s formula 
and have been tabulated up to Gin Chappell’s Interpolation Tables. The required 
functions of a, b, c, .. . are easily obtained as the even differences of the symmetrical 
series . . . chaabe.... 
The Jordan formula is therefore of an interesting new type which does not require 
the previous preparation of differences, and it is well adapted to routine computation. 
It is not claimed that an interpolate is obtained as quickly with the formula as by 
using Everett’s formula with even differences already provided. The Committee gave 
some consideration to this point, and finally decided that for the tables in the volume 
now being published, which are all single-entry tables, the expense of printing the 
necessary number of even order differences was fully justified. The formula is worthy 
of consideration in the case of tables not provided with differences. 
It should be mentioned that interpolation in a table not provided with differences 
may be performed more quickly by the aid of the Lagrange formula, given tables of 
the coefficients and a calculating machine of sufficient capacity. The general use of 
Lagrange’s formula is, however, not convenient for three reasons: (1) No tables of 
the coefficients have been published, although the Committee has manuscript tables 
of the 4-point and 6-point coefficients at intervals of 0-001; (2) the tables required 
are more extensive than those for Everett’s formula, because separate tables must 
be used according to the number of tabular entries employed ; (3) it is not possible 
to tell by inspection of a table how many values should be used in order to interpolate 
to the full accuracy of the table. 
In connection with the expense of printing differences a further consideration is 
that the fourth, sixth, eighth and tenth differences, if less than 1000, 10,000, 100,000 
and 1,000,000 respectively, can be dispensed with entirely if they are multiplied by 
the factors —0:18393, —0-20697, —0-21803 and —0-22456 respectively, and the 
products added to the difference of the next lower even order, which is then used in 
exactly the same way as an unmodified difference. This method of ‘‘throw back” 
has been adopted where applicable in the volume prepared, resulting in the saving 
of six pages of printing. In each case where it is applied (and this will be indicated) 
the user is saved the trouble of finding two coefficients from the tables of coefficients, 
and of performing two multiplications. The modification of the differences, however, 
leads to more labour on the part of the proof-reader, who is responsible for the accuracy 
of every printed column. It also complicates another problem that sometimes 
arises—namely that of finding differentials from the differences. 
In cases where the functions tabulated are related in such a way that they are 
successive derivatives of one function, no differences are needed, except perhaps at 
one end of the series of functions, as Taylor’s theorem can be applied directly. An 
instance of this will be found in the Hh functions (integrals of the probability integral) 
in the forthcoming volume. 
The Committee is indebted to Dr. A. C. Aitken, of the Mathematical Department 
of the University of Edinburgh, for help given in investigating the Jordan formula. 
A paper on “Interpolation without Differences ” is being presented to the Mathematics 
Sub-section of Section A by Dr. J. Wishart (see p. 17). 
Future programme.—tThe programme outlined in last year’s report is still before 
the Committee, but each of the propositions there put forward will require considerable 
financial assistance. The Committee would like to be able to publish in the near 
future the tables of elliptic functions (described in the last report) computed for the 
Association under the superintendence of the late J. W. L. Glaisher. For the purpose 
of checking these by differencing (the original 10-figure calculations are in the 
possession of the Committee in the form of thirty bound manuscript volumes, but 
exhaustive inquiry has failed to reveal any trace of systematic checking, especially 
by differencing) and preparing the manuscript of 200 pages for the printer, a sum of 
£100 would be required. The printing would cost about £450—£500. 
A request has been received from Prof. S. Chapman for the calculation of the 
associated Legendre functions P,”(v) and Q,™ (vy) for values 0, 1 and 2 of m and 
values of n that are half an odd integer. These are required for numerical work in 
connection with anchor ring problems. 
Reappoiniment.—The Committee desires to be reappointed, with a grant for 
general purposes of £100, which would be expended on the tables of the associated 
Legendre functions and on work for the proposed volumes of Bessel and confluent 
hypergeometric functions. 
